The eigenvalue problem for a coupled system of singular p-Laplacian differential equations involving fractional differential-integral conditions

In this paper, we deal with a coupled system of singular p-Laplacian differential equations involving fractional differential-integral conditions. By employing Schauder’s fixed point theorem and the upper and lower solution method, we establish an eigenvalue interval for the existence of positive solutions. As an application an example is presented to illustrate the main results.

where \(D^{\alpha_{i}},D^{\beta_{i}}, D^{\gamma_{i}}\) (\(i=1,2\)) are the standard Riemannn-Liouville fractional derivatives, \(I^{\omega_{i}}\) is the Riemannn-Liouville fractional integral, \(\varphi_{p_{i}}\) is the p-Laplacian operator defined by \(\varphi_{p_{i}}(s)=|s|^{p_{i}-2}s,p_{i}>2 \) (\(i=1,2\)), and the nonlinearity \(f_{1}(x,y,z)\) may be singular at \(x=0,y=0,z=0\).

Let \(q_{i}\) satisfies the relation \(\frac{1}{q_{i}}+\frac {1}{p_{i}}=1\), where \(p_{i}\) is given by (1.1), then \(1< q_{i}<2\).

Fractional calculus provides an excellent tool for describing the hereditary properties of various materials and processes. Concerning the development of theory, method and application of fractional calculus, we refer the reader to the recent papers [1–8].

On the other hand, the study of coupled systems involving fractional differential equations is also important as such systems occur in various problems of applied nature. So considerable work has been done to study the existence result for them nowadays [9–12]. The authors got the existence solutions by the method of the fixed point theorem, the coincidence degree theorem, or Schauder’s fixed point theorem.

The theory of upper and lower solutions is well known to be an effective method to deal with the existence of solutions for the boundary value problems of the fractional differential equations. In [13] the authors used the method of upper and lower solutions and investigated the existence of solutions for initial value problems. By the same method some people got the solutions of boundary value problems for fractional differential equations, such as [14, 15]. To the best of our knowledge, only few papers considered the existence of solutions by using the method of upper and lower solutions for boundary value problems with fractional coupled systems.

The aim of this paper is to deal with the eigenvalue problem for a coupled system of fractional differential equations involving differential-integral conditions. The novelty of this paper is that the nonlinear terms \(f_{1}\), \(f_{2}\) in the system (1.1) involve different unknown functions \(u_{1}(t)\), \(u_{2}(t)\) and their Riemann-Liouville fractional derivatives with different orders, and \(f_{1}(x,y,z)\) may be singular at \(x=0, y=0, z=0\). We establish an eigenvalue interval for the existence of positive solutions by Schauder’s fixed point theorem and the upper and lower solutions method.

Lemma 2.5

Let\(u_{i}(t)=I^{\gamma_{i}}v_{i}(t)\), \(v_{i}(t)\in C[0,1]\) (\(i=1,2\)). Then (1.1) can be transformed into (2.3). Moveover, if\((v_{1}(t),v_{2}(t))\in C[0,1]\times C[0,1]\)is a positive solution of the problem (2.3), then\((I^{\gamma_{1}}v_{1}(t), I^{\gamma_{2}}v_{2}(t))\)is a positive solution of the problem (1.1).

Proof

Let \(u_{i}(t)=I^{\gamma_{i}}v_{i}(t),v_{i}(t)\in C[0,1]\), by the definition of the Riemannn-Liouville fractional derivatives and integrals, we obtain

Consequently, if \((v_{1}(t),v_{2}(t))\in C[0,1]\times C[0,1]\) is a positive solution of the problem (2.3), then \((I^{\gamma_{1}}v_{1}(t), I^{\gamma_{2}}v_{2}(t))\) is a positive solution of the problem (1.1).

It is well know that \((v_{1},v_{2})\in C[0,1]\times C[0,1]\) is a solution of system (2.3), if and only if \((v_{1},v_{2})\in C[0,1]\times \in C[0,1]\) is a solution of the following nonlinear integral equation system:

Proof

To establish the existence of a solution to the boundary value problem (1.1), we need to make the following assumptions.

(H1):

\(f_{1}(x,y,z): (0,+\infty)^{3}\rightarrow[0,+\infty]\) is continuous and non-increasing in \(x,y,z>0\), respectively, and for all \(r\in(0,1)\), there exists a constant \(\varepsilon>0\), such that, for any \((x,y,z)\in(0,+\infty)^{3}\), we have

$$f_{1}(rx,ry,rz)\leq r^{-\varepsilon}f_{1}(x,y,z). $$

(H2):

\(f_{2}(t,x): [0,1]\times[0,+\infty)\rightarrow[0,+\infty]\) is continuous and non-decreasing in \(x>0\), and there exists a constant \(0<\sigma<\frac{1}{q_{2}-1}\), such that, for any \(r\in(0,1)\), \((t,x)\in[0,1]\times[0,+\infty)\), we have

Then there exists a constant\(\lambda^{*}>0\)such that for any\(\lambda \in(\lambda^{*},+\infty)\), the BVP (1.1) has at least one positive solution\((u_{1}(t),u_{2}(t))\), and, moreover, there exist two constants\(0< l<1\)and\(L>1\)such that

We assert that \(T_{\lambda}\) is well defined and \(T_{\lambda}(P)\subset P\). In fact, for any \(v_{1}(t)\in P\), there exists a positive number \(0< l_{v_{1}}<1\) such that \(l_{v_{1}}t^{\alpha_{1}-\gamma_{1}-1}\leq v_{1}(t)\leq l_{v_{1}}^{-1}t^{\alpha_{1}-\gamma_{1}-1}, t\in[0,1]\). It follows from Lemma 2.3 and (H2) that

This implies that \(T_{\lambda}\) is well defined and \(T_{\lambda}(P)\subset P\). Furthermore, comparing (3.3) and (2.2), the right hand side of (3.3) is exactly the same as the right hand of (2.2), if \(h_{1}(t)\) in (2.1) is taken as \(\lambda f_{1}(I^{\gamma_{1}}v_{1}(t),v_{1}(t),Av_{1}(t))\). Hence as the left hand side of (2.2), i.e. \(v_{1}(t)\) satisfies equation (2.1) according to Lemma 2.4, the left hand side of (3.3), i.e. \(T_{\lambda}v_{1}(t)\) must also satisfy equation (2.1) with \(h_{1}(t)\) replace by \(\lambda f_{1}(I^{\gamma_{1}}v_{1}(t),v_{1}(t),Av_{1}(t))\), namely

Then \(D_{\lambda^{*}}:C[0,1]\rightarrow C[0,1]\), and a fixed point of the operator \(D_{\lambda^{*}}\) is a solution of the BVP (3.23). On the other hand, from the definition of F and the fact that the function \(f_{1}(x,y,z)\) is non-increasing in \(x, y, z\) respectively, and A is non-decreasing, we obtain \(f_{1}(I^{\gamma_{1}}\Phi(t),\Phi(t),A\Phi(t)) \leq F(v_{1}(t))\leq f_{1}(I^{\gamma_{1}}\Psi(t),\Psi(t),A\Psi(t))\), provided that \(\Psi(t)\leq v_{1}(t)\leq\Phi(t)\), \(F(v_{1}(t))=f_{1}(I^{\gamma_{1}}\Psi(t), \Psi(t),A\Psi (t))\), provided that \(v_{1}(t)<\Psi(t)\), and \(F(v_{1}(t))=f_{1}(I^{\gamma_{1}}\Phi(t),\Phi(t),A\Phi(t))\), provided that \(v_{1}(t)>\Phi(t)\). So we have

On the other hand, let \(\Omega\subset E\) be bounded. As the function \(H_{1}(t,s), G(\beta_{1},t,s)\) is uniformly continuous on \([0,1]\times[0,1]\), \(D_{\lambda^{*}}(\Omega)\) is equicontinuous. By the Arzela-Ascoli theorem, we have \(D_{\lambda ^{*}}:E\rightarrow E\) is completely continuous. Thus by using the Schauder fixed point theorem, \(D_{\lambda^{*}}\) has at least one fixed point x such the \(x=D_{\lambda^{*}}x\).

Now we prove

$$\Psi(t)\leq x(t)\leq\Phi(t), \quad t\in[0,1]. $$

Since x is a fixed point of \(D_{\lambda^{*}}\), by (3.18) and (3.23), we have

Hence, (H3) holds. Then by Theorem 3.1 there exists a constant \(\lambda^{*}>0\) such that for any \(\lambda\in (\lambda^{*},+\infty)\), the BVP (1.1) has at least one positive solution \((u_{1}(t),u_{2}(t))\).

Acknowledgements

The author sincerely thanks the editor and reviewers for their valuable suggestions and useful comments to improve the manuscript.

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